# Joint budget for the four transition pieces **The sufficient twin margin remains OPEN, and nothing below estimates it.** The outside remainder E_out has no derived bound at all here beyond the tautology inherited from S(x)>=0. The one analytic attempt made below does not improve the joint budget; it is a scoped negative for a named mechanism, and its own decisive input, the norm lower bound of [sharp-corner-transition.md](sharp-corner-transition.md) section 4, is retained at its stated scope by transition-round-audit.md. No novelty is claimed. Baseline: mathematical commit `0292082`, working checkout at `cb51292` (clean at launch). Definitions, cutoffs and equations (E1), (E2) are those of [TRANSITION-RESEARCH-EXECUTION.md](TRANSITION-RESEARCH-EXECUTION.md) section 2. Owning derivations: [corner-correlation.md](corner-correlation.md), [corner-coefficient-energy.md](corner-coefficient-energy.md), [sharp-corner-transition.md](sharp-corner-transition.md). Consumer: [RESEARCH-HANDOFF.md](RESEARCH-HANDOFF.md) section 3 and [endpoint-target-audit.md](endpoint-target-audit.md). ## 1. The decomposition, its supports, and what it does not touch ### 1.1 The split is exact and pointwise With rho_i and h_i=1-rho_i as in the contract, for every m, \[ 1_{d>D_i}-h_i(d)=\rho_i(d)1_{D_i=z_i, and rho_i(d)=1-h_i(d) on the transition band. Hence, on the interval I_i where the upper divisor cut is redundant, \[ T_i(m)=C_i(m)-\widetilde C_i(m) =\sum_{\substack{d\mid m\\ D_iW_i with r | m/d, so m/d>=W_i+1 and d<=m/(W_i+1). On I_L=J_x this gives d<=floor(x/(V+1)) =D_L^+, and on I_R=(x/2-2,x-2] it gives d<=floor((x-2)/(Z+1))=D_R^+. So the upper cut may be written or omitted in C_i, \widetilde C_i and T_i without changing any of the four sums. The finite check in section 6 also exhibits terms outside I_L for which the cut is **not** redundant, so the hypothesis is not treated as free. Both h_i and rho_i 1_{(D_i,z_i)} vanish for d<=D_i, so every one of the four pieces is supported on the same divisor rectangle D_Lx^{19/25-2eta} and e>x^{19/20-2eta}, so \[ de>x^{171/100-4\eta},\quad d^5e^2>x^{57/10-14\eta},\quad de^3>x^{361/100-8\eta}. \] Against floor(x^{3/4}), floor(x^{49/20}) and floor(x^{321/200}) these hold for all sufficiently large x when eta<6/25, eta<13/56 and eta<401/1600 respectively. The third W_dagger condition is a disjunction, and its **second** disjunct de^3>floor(x^{321/200}) is satisfied with a margin of about x^2. [corner-correlation.md](corner-correlation.md) section 0 verifies the inclusion through the first disjunct d>floor(x^{151/200}), which needs eta<1/400. That check is correct, but the inclusion S_0 subset W_dagger is not what forces eta<1/400: at fixed eta the binding constraints are elsewhere, namely etaS}\,dS/S. Write, for D_i<=S<=z_i, \[ C_i^{(S)}(m)=\sum_{\substack{d\mid m\\ d>S}}\mu(d)\beta_{W_i}(m/d), \qquad G_i^{(S)}(m)=\sum_{\substack{d\mid m\\ D_i_{S,T}, then apply Cauchy in n at each fixed (S,T). The signed quantity actually bounded at fixed (S,T) is the short-window sharp corner correlation sum_n G_L^{(S)}(n)G_R^{(T)}(n-2), with divisor windows (D_L,S] and (D_R,T], both shifted endpoints, all prime powers r inside beta with weight Lambda(r)=log p, the (d,e)|2 compatibility and both gcd branches retained. Two operations discard cancellation: the triangle inequality on the (S,T) average discards sign variation in the cutoffs, and Cauchy in n discards the shift-2 sign structure. ### 3.3 The cost: the route cannot beat the direct Cauchy bound Write nu_L(S)=||G_L^{(S)}||_{I_0} and nu_R(T)=||G_R^{(T)}||_{I_2}. The mechanism outputs \[ \mathcal M=\langle\nu_L(S)\rangle_S\cdot\langle\nu_R(T)\rangle_T . \] Both averages are against probability measures, so Minkowski's integral inequality applied to (5) gives ||T_L||<=_S and ||T_R||<=_T, hence \[ \mathcal M\ \ge\ \|T_L\|\cdot\|T_R\| . \tag{7} \] Inequality (7) is unconditional. It compares the exact norm output of this average-triangle-Cauchy procedure with the exact product of transition norms. It does not compare arbitrary upper constants, and does not apply to an argument that retains signed cancellation before these inequalities. The finite check in section 6 confirms the direction and finds it strict on the proxies, so (7) is not an equality being read as a loss. Combining (7) with the derived lower bound ||T_i||^2>=(L_0w_i eta/64)x log^2 x, valid for sufficiently large x at each fixed 0=1**, \[ \Sigma_{(D,S]}(N)=\sum_{m\le N}\Big(\sum_{d\mid m,\ D=(L_0 w_i eta/64)x log^2 x gives \[ \liminf_{x\to\infty}\frac1{L_i}\int_{D_i}^{z_i} \sqrt{\theta_{\rm eff}(S)}\,\frac{dS}{S} \ \ge\sqrt{L_0w_i/128}>0. \tag{9} \] This is along the fixed-eta dyadic sequence, for each side. The bounded cap makes it legitimate to exclude uniform o(1) outside a vanishing logarithmic proportion: a bounded exceptional part then has vanishing integral too. Without boundedness or uniform integrability that inference would not follow. The exact constants in the two sides' L_i retain floors; L_i=eta log x+o(1) is used only in taking the limit. A bound valid only on a restricted N-range must be combined with a bound for the omitted part. The crude upper budget there does not establish its actual size or make every restricted-range theorem unusable. Nor does (9) preclude localized improvements; it constrains this averaged norm procedure. ### 3.5 What the reformulation does leave Two consequences of (6) are worth keeping, since they replace three separate norm bounds by one quantity and one obligation. First, because the averages in (6) are against probability measures, \[ |R_{10}+R_{01}+R_{11}|\ \le\ \sup_{\substack{D_L\le S\le z_L\\ D_R\le T\le z_R}} \big|\Phi(D_L,D_R)-\Phi(S,T)\big| . \tag{10} \] A modulus of continuity for Phi in its two cutoff parameters bounds the whole non-smoothed part. Present knowledge supplies only the trivial sup<=2sup|Phi|, which returns the budget of section 2.1. Second, R_cor=R_00+(R_cor-R_00), so with (10), \[ |R_{\rm cor}|\ \le\ |R_{00}|+\sup|\Phi(D_L,D_R)-\Phi(S,T)| . \tag{11} \] In the particular triangle bound (11), improving the continuity term alone leaves the available 2Ax log x bound for |R_00|. That separate budget is insufficient. It does not follow that every grouping requires an individual R_00 estimate: the terms can cancel before taking absolute values. Its coefficient 1 in (2) proves no such necessity. If one chooses to require |R_00|=o(x) by a fixed-power logarithmic saving from x log x, a saving exponent greater than one would suffice. That is one sufficient target, not a necessary global proof obligation. A suitable constant-sized one-sided O(x) bound may already fit an allocated budget; a joint estimate may bypass the individual term altogether. ## 4. Sufficient consumer, stated with every term The template of the assignment is B_mix=R_00+R_10+R_01 and, on a common unbounded set of dyadic x, \[ R_{11}\ge-F_{11},\quad B_{\rm mix}\ge-F_{\rm mix},\quad E_{\rm out}\ge-F_{\rm out},\qquad F_{11}+F_{\rm mix}+F_{\rm out}\le C_2x-2cx/\log^Kx, \] with fixed c,K>0 and fixed H>K in the remaining error. What the present state actually supplies: | term | sign | scale set | best available F | status | F/x | |---|---|---|---|---|---| | R_11 | unknown | every sufficiently large dyadic x, fixed eta=0, namely E_out>=-C_2x-R_cor-O_H(x/log^Hx), which is the tautology identified in [corner-correlation.md](corner-correlation.md) section 2.2 and carries no information. **Absence of an estimate for E_out is not evidence that the corner method fails**, and nothing here converts it into one. A rescaled-average or cumulative consumer is equally admissible (endpoint-target-audit sections 2 and 3, consumer-comparison). The four budgets above are uniform in x, so they average to the same orders; no sampling or exceptional-set argument improves them. Averaging is therefore not a way around section 2.1 for these pieces, though it may be for a future signed estimate. ## 5. Reusable statements and limits Retain the exact split (1)–(2), the cutoff averages (5), the mixed finite difference (6), the norm-output comparison (7), its small-fixed-eta saturation (8), the corrected bounded-average constraint (9), and the conditional modulus bounds (10)–(11). None supplies a signed estimate. The complete joint consumer in section 4, or an admissible averaged alternative, remains the criterion for a proof payoff. ## 6. Verification and falsifiers [transition-joint-budget-validation.js](transition-joint-budget-validation.js) checks, on two finite fixtures with proxy cutoffs and both shifted endpoints: the pointwise split C_i=T_i+\widetilde C_i; the four-piece identity (2); the reconstruction of R_11 from the determinant-2 kernel (E1) with the (d,e)|2 test and the gcd 2 branch present; the redundancy of the upper divisor cut on the interval, together with witnesses off the interval where it fails; the moving-cutoff representation (5), computed by exact piecewise integration over integer breakpoints; the cutoff averages and the mixed second difference (6); and the direction of the Minkowski step (7). Eight controls are active and each changes the answer: exchanging rho and h, using the complementary window (S,z] instead of (D,S], replacing Lambda(r) by log r, dropping the lower cut in the smoothed weight, evaluating the right factor at n instead of n-2, dropping the compatibility test, using the upper cut off its interval, and reading (7) as an equality. These are finite identity checks with proxy cutoffs. They certify no asymptotic norm order, no sign, no saving and no twin margin. The imported mean squares and the derived lower bound (16) are not tested by them. ## 7. Research use The norm obstruction applies only after the stated inequalities lose the signs. A signed estimate for the cutoff correlation or its joint average remains open. A lower smoothed norm or a local window refinement could be supporting information, but neither is a required prerequisite for a joint proof. See [transition-round-audit.md](transition-round-audit.md) for the scope corrections and finite countermodels. No region, exact cut or sufficient twin margin has changed.